metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.22D14, Q8⋊C4⋊1D7, (C8×Dic7)⋊22C2, Q8⋊Dic7⋊2C2, (C2×C8).207D14, (C2×Q8).14D14, C28.3Q8⋊5C2, C14.D8.3C2, C2.D56.8C2, C28.18(C4○D4), C14.68(C4○D8), C4.31(C4○D28), (C2×Dic7).92D4, C22.194(D4×D7), C2.7(Q8.D14), C4.57(D4⋊2D7), (C2×C28).240C23, (C2×C56).195C22, C28.23D4.4C2, (C2×D28).59C22, C4⋊Dic7.88C22, (Q8×C14).23C22, C14.29(C4.4D4), C2.16(SD16⋊3D7), C7⋊3(C42.78C22), (C4×Dic7).227C22, C2.19(Dic7.D4), (C7×Q8⋊C4)⋊17C2, (C2×C14).253(C2×D4), (C7×C4⋊C4).41C22, (C2×C7⋊C8).217C22, (C2×C4).347(C22×D7), SmallGroup(448,334)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊Dic7⋊C2
G = < a,b,c,d | a2=b8=1, c14=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=ab4c13 >
Subgroups: 532 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C42.C2, C7⋊C8, C56, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C42.78C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×D28, Q8×C14, C14.D8, C8×Dic7, C2.D56, Q8⋊Dic7, C7×Q8⋊C4, C28.3Q8, C28.23D4, Q8⋊Dic7⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C4○D8, C22×D7, C42.78C22, C4○D28, D4×D7, D4⋊2D7, Dic7.D4, SD16⋊3D7, Q8.D14, Q8⋊Dic7⋊C2
►On 224 points
Generators in S224
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 178 56 90 211 64 164 129)(2 105 165 193 212 116 29 79)(3 180 30 92 213 66 166 131)(4 107 167 195 214 118 31 81)(5 182 32 94 215 68 168 133)(6 109 141 169 216 120 33 83)(7 184 34 96 217 70 142 135)(8 111 143 171 218 122 35 57)(9 186 36 98 219 72 144 137)(10 85 145 173 220 124 37 59)(11 188 38 100 221 74 146 139)(12 87 147 175 222 126 39 61)(13 190 40 102 223 76 148 113)(14 89 149 177 224 128 41 63)(15 192 42 104 197 78 150 115)(16 91 151 179 198 130 43 65)(17 194 44 106 199 80 152 117)(18 93 153 181 200 132 45 67)(19 196 46 108 201 82 154 119)(20 95 155 183 202 134 47 69)(21 170 48 110 203 84 156 121)(22 97 157 185 204 136 49 71)(23 172 50 112 205 58 158 123)(24 99 159 187 206 138 51 73)(25 174 52 86 207 60 160 125)(26 101 161 189 208 140 53 75)(27 176 54 88 209 62 162 127)(28 103 163 191 210 114 55 77)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 135 197 110)(2 95 198 120)(3 133 199 108)(4 93 200 118)(5 131 201 106)(6 91 202 116)(7 129 203 104)(8 89 204 114)(9 127 205 102)(10 87 206 140)(11 125 207 100)(12 85 208 138)(13 123 209 98)(14 111 210 136)(15 121 211 96)(16 109 212 134)(17 119 213 94)(18 107 214 132)(19 117 215 92)(20 105 216 130)(21 115 217 90)(22 103 218 128)(23 113 219 88)(24 101 220 126)(25 139 221 86)(26 99 222 124)(27 137 223 112)(28 97 224 122)(29 183 151 83)(30 68 152 196)(31 181 153 81)(32 66 154 194)(33 179 155 79)(34 64 156 192)(35 177 157 77)(36 62 158 190)(37 175 159 75)(38 60 160 188)(39 173 161 73)(40 58 162 186)(41 171 163 71)(42 84 164 184)(43 169 165 69)(44 82 166 182)(45 195 167 67)(46 80 168 180)(47 193 141 65)(48 78 142 178)(49 191 143 63)(50 76 144 176)(51 189 145 61)(52 74 146 174)(53 187 147 59)(54 72 148 172)(55 185 149 57)(56 70 150 170)
G:=sub<Sym(224)| (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,178,56,90,211,64,164,129)(2,105,165,193,212,116,29,79)(3,180,30,92,213,66,166,131)(4,107,167,195,214,118,31,81)(5,182,32,94,215,68,168,133)(6,109,141,169,216,120,33,83)(7,184,34,96,217,70,142,135)(8,111,143,171,218,122,35,57)(9,186,36,98,219,72,144,137)(10,85,145,173,220,124,37,59)(11,188,38,100,221,74,146,139)(12,87,147,175,222,126,39,61)(13,190,40,102,223,76,148,113)(14,89,149,177,224,128,41,63)(15,192,42,104,197,78,150,115)(16,91,151,179,198,130,43,65)(17,194,44,106,199,80,152,117)(18,93,153,181,200,132,45,67)(19,196,46,108,201,82,154,119)(20,95,155,183,202,134,47,69)(21,170,48,110,203,84,156,121)(22,97,157,185,204,136,49,71)(23,172,50,112,205,58,158,123)(24,99,159,187,206,138,51,73)(25,174,52,86,207,60,160,125)(26,101,161,189,208,140,53,75)(27,176,54,88,209,62,162,127)(28,103,163,191,210,114,55,77), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,135,197,110)(2,95,198,120)(3,133,199,108)(4,93,200,118)(5,131,201,106)(6,91,202,116)(7,129,203,104)(8,89,204,114)(9,127,205,102)(10,87,206,140)(11,125,207,100)(12,85,208,138)(13,123,209,98)(14,111,210,136)(15,121,211,96)(16,109,212,134)(17,119,213,94)(18,107,214,132)(19,117,215,92)(20,105,216,130)(21,115,217,90)(22,103,218,128)(23,113,219,88)(24,101,220,126)(25,139,221,86)(26,99,222,124)(27,137,223,112)(28,97,224,122)(29,183,151,83)(30,68,152,196)(31,181,153,81)(32,66,154,194)(33,179,155,79)(34,64,156,192)(35,177,157,77)(36,62,158,190)(37,175,159,75)(38,60,160,188)(39,173,161,73)(40,58,162,186)(41,171,163,71)(42,84,164,184)(43,169,165,69)(44,82,166,182)(45,195,167,67)(46,80,168,180)(47,193,141,65)(48,78,142,178)(49,191,143,63)(50,76,144,176)(51,189,145,61)(52,74,146,174)(53,187,147,59)(54,72,148,172)(55,185,149,57)(56,70,150,170)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,178,56,90,211,64,164,129)(2,105,165,193,212,116,29,79)(3,180,30,92,213,66,166,131)(4,107,167,195,214,118,31,81)(5,182,32,94,215,68,168,133)(6,109,141,169,216,120,33,83)(7,184,34,96,217,70,142,135)(8,111,143,171,218,122,35,57)(9,186,36,98,219,72,144,137)(10,85,145,173,220,124,37,59)(11,188,38,100,221,74,146,139)(12,87,147,175,222,126,39,61)(13,190,40,102,223,76,148,113)(14,89,149,177,224,128,41,63)(15,192,42,104,197,78,150,115)(16,91,151,179,198,130,43,65)(17,194,44,106,199,80,152,117)(18,93,153,181,200,132,45,67)(19,196,46,108,201,82,154,119)(20,95,155,183,202,134,47,69)(21,170,48,110,203,84,156,121)(22,97,157,185,204,136,49,71)(23,172,50,112,205,58,158,123)(24,99,159,187,206,138,51,73)(25,174,52,86,207,60,160,125)(26,101,161,189,208,140,53,75)(27,176,54,88,209,62,162,127)(28,103,163,191,210,114,55,77), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,135,197,110)(2,95,198,120)(3,133,199,108)(4,93,200,118)(5,131,201,106)(6,91,202,116)(7,129,203,104)(8,89,204,114)(9,127,205,102)(10,87,206,140)(11,125,207,100)(12,85,208,138)(13,123,209,98)(14,111,210,136)(15,121,211,96)(16,109,212,134)(17,119,213,94)(18,107,214,132)(19,117,215,92)(20,105,216,130)(21,115,217,90)(22,103,218,128)(23,113,219,88)(24,101,220,126)(25,139,221,86)(26,99,222,124)(27,137,223,112)(28,97,224,122)(29,183,151,83)(30,68,152,196)(31,181,153,81)(32,66,154,194)(33,179,155,79)(34,64,156,192)(35,177,157,77)(36,62,158,190)(37,175,159,75)(38,60,160,188)(39,173,161,73)(40,58,162,186)(41,171,163,71)(42,84,164,184)(43,169,165,69)(44,82,166,182)(45,195,167,67)(46,80,168,180)(47,193,141,65)(48,78,142,178)(49,191,143,63)(50,76,144,176)(51,189,145,61)(52,74,146,174)(53,187,147,59)(54,72,148,172)(55,185,149,57)(56,70,150,170) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,178,56,90,211,64,164,129),(2,105,165,193,212,116,29,79),(3,180,30,92,213,66,166,131),(4,107,167,195,214,118,31,81),(5,182,32,94,215,68,168,133),(6,109,141,169,216,120,33,83),(7,184,34,96,217,70,142,135),(8,111,143,171,218,122,35,57),(9,186,36,98,219,72,144,137),(10,85,145,173,220,124,37,59),(11,188,38,100,221,74,146,139),(12,87,147,175,222,126,39,61),(13,190,40,102,223,76,148,113),(14,89,149,177,224,128,41,63),(15,192,42,104,197,78,150,115),(16,91,151,179,198,130,43,65),(17,194,44,106,199,80,152,117),(18,93,153,181,200,132,45,67),(19,196,46,108,201,82,154,119),(20,95,155,183,202,134,47,69),(21,170,48,110,203,84,156,121),(22,97,157,185,204,136,49,71),(23,172,50,112,205,58,158,123),(24,99,159,187,206,138,51,73),(25,174,52,86,207,60,160,125),(26,101,161,189,208,140,53,75),(27,176,54,88,209,62,162,127),(28,103,163,191,210,114,55,77)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,135,197,110),(2,95,198,120),(3,133,199,108),(4,93,200,118),(5,131,201,106),(6,91,202,116),(7,129,203,104),(8,89,204,114),(9,127,205,102),(10,87,206,140),(11,125,207,100),(12,85,208,138),(13,123,209,98),(14,111,210,136),(15,121,211,96),(16,109,212,134),(17,119,213,94),(18,107,214,132),(19,117,215,92),(20,105,216,130),(21,115,217,90),(22,103,218,128),(23,113,219,88),(24,101,220,126),(25,139,221,86),(26,99,222,124),(27,137,223,112),(28,97,224,122),(29,183,151,83),(30,68,152,196),(31,181,153,81),(32,66,154,194),(33,179,155,79),(34,64,156,192),(35,177,157,77),(36,62,158,190),(37,175,159,75),(38,60,160,188),(39,173,161,73),(40,58,162,186),(41,171,163,71),(42,84,164,184),(43,169,165,69),(44,82,166,182),(45,195,167,67),(46,80,168,180),(47,193,141,65),(48,78,142,178),(49,191,143,63),(50,76,144,176),(51,189,145,61),(52,74,146,174),(53,187,147,59),(54,72,148,172),(55,185,149,57),(56,70,150,170)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 2 | 2 | 8 | 8 | 14 | 14 | 14 | 14 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C4○D28 | D4⋊2D7 | D4×D7 | SD16⋊3D7 | Q8.D14 |
| kernel | Q8⋊Dic7⋊C2 | C14.D8 | C8×Dic7 | C2.D56 | Q8⋊Dic7 | C7×Q8⋊C4 | C28.3Q8 | C28.23D4 | C2×Dic7 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 3 | 3 | 8 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Q8⋊Dic7⋊C2 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 51 | 0 | 0 |
| 0 | 0 | 51 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 82 |
| 0 | 0 | 0 | 0 | 31 | 31 |
| 34 | 10 | 0 | 0 | 0 | 0 |
| 79 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 62 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 13 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 26 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 51 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 82 | 31 |
| 0 | 0 | 0 | 0 | 31 | 31 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,51,0,0,0,0,51,15,0,0,0,0,0,0,31,31,0,0,0,0,82,31],[34,79,0,0,0,0,10,0,0,0,0,0,0,0,15,62,0,0,0,0,0,98,0,0,0,0,0,0,100,13,0,0,0,0,13,13],[112,26,0,0,0,0,0,1,0,0,0,0,0,0,98,0,0,0,0,0,51,15,0,0,0,0,0,0,82,31,0,0,0,0,31,31] >;
Q8⋊Dic7⋊C2 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_7\rtimes C_2 % in TeX
G:=Group("Q8:Dic7:C2"); // GroupNames label
G:=SmallGroup(448,334);
// by ID
G=gap.SmallGroup(448,334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,120,1094,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^14=a,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^3,d*c*d^-1=a*b^4*c^13>;
// generators/relations